The formation of the equations relative to the differential coefficients of higher orders can present no difficulty. (44.) All these equations, and the equation u = 0, would be verified; that is, that in each, the left side would become identical with the right side, if we were to substitute for y the function of r, it represents, and for the differential coefficients of y, the differential coefficients of that function. This is expressed by saying, that these various equations subsist or obtain together. Hence, by combining them in any way whatever, other equations will be formed, which will subsist or obtain with them.

(45.) An equation which contains one or several differential coefficients is called a differential equation; and
a primitive equation is that which does not contain any.

A differential equation of the first order is that which contains no other differential coefficient than the first,
and generally it is said to be of the nth order, when the n" differential coefficient is the highest it contains.

The degree of a differential equation is the highest power of the differential coefficient, which marks its order,
it contains. Thus a differential equation of the nth order in which the highest power of the n" differential
coefficient would be the m", would be of the m" degree.

(46.) From the remark we have made in (44), we already perceive that several differential equations of the same order may correspond to the same primitive equation. Thus, it is obvious that from each combination of a differential equation of the mth order, with the differential equations of the preceding orders, will result another differential equation of the mth order. But among the various differential equations of the same order which may be so obtained, some require a peculiar attention, because they express more general relations between x, y, and the differential coefficients of y, than the others.

We must first observe, that by differentiating a primitive equation between r and y, that is, by applying the

dy
rule given (42) to form the equation which gives the value of it may happen that one of the constants con-
dx'

tained in the equation should disappear. It would obviously be the case, for instance, with respect to the con-
stant a, if the primitive equation had the form ƒ (x, y) = a; and if a were not contained in f(x, y). But in
all cases, by combining the primitive equation with the differential equation of the first order, so as to eliminate
one of the constants, it will always be easy to obtain a differential equation of the first order, containing one
constant less than the primitive equation.

Such a differential equation does not only correspond to the proposed primitive equation, but to all those dy which differ from it by the value of the constant. Hence it expresses a relation between x, y, and more dr

general, than a differential equation of the first order containing that constant.

If the constant eliminated enter the primitive equation in a degree higher than the first, the result to which we shall arrive will contain the differential coefficient of the first order in a degree higher than the first.

(47.) These considerations may easily be extended to differential equations of higher orders. We shall be able, for instance, to eliminate two constants between the primitive equation, the differential equation of the first order, and that upon which depends the value of the differential coefficient of the second order; and the result will be a differential equation of the second order containing two constants less than the primitive equation. Generally, we see that we may obtain a differential equation of the m order, containing m constants less than the primitive equation.

(48.) Instead of eliminating constants between the primitive equation, and its differential equations, they might be combined so as to make other quantities disappear in the result. The variables x or y, for instance, or any function of them entering the primitive and differential equations might be eliminated.

We shall now apply the foregoing rules and observations relative to implicit functions of x, or to equations between the two variables r and y, to a few examples.

Example 1. Let it be proposed to determine the values of the first and second differential coefficients of the implicit function of x which y represents in the equation

To find the second differential coefficient, we shall take the differential coefficient of both sides of (a), consi

{ 6 b x (3 a y2 - cx)2 + 6 a y (3 b x2 − c y)2 - 2 c (3 b x2 - cy) (3 a y2 — c x) } ̧
(3 ay2 - cx)3

In this case the primitive equation, containing no higher power of y than the second, may be resolved with respect to that variable. It gives

Part L.

It is easy to verify that the two values we have thus obtained for the differential coefficient of y are identical with those we might derive from the value of y.

There is still another manner in which we might arrive at the value of dy, expressed in terms of ♬ alone.

x

We might eliminate y between the primitive equation, and the differential equation of the first order, by taking the value of y in the last, where it enters only in the first degree, and substituting it in the other, we shall have, by this process, the following equation,

and by eliminating a we find for differential equation of the first order independent of a

Differentiating this equation, b will disappear, and we shall obtain a differential equation of the second order independent of the two constants a and b,

It is easy to verify that this equation is satisfied by the value of y given by the primitive equation. For this

tuted in the differential equation of the second order, makes one side identically equal to the other.

taking the value of a in that equation, and substituting it in the primitive equation, we shall have

We may now substitute in this equation, instead of (a2 + x)", its value taken in the primitive equation, and then we shall obtain a differential equation independent of that irrational function of x,

Example 7. We shall take for the last example an equation containing logarithmic, exponential, and trigono metrical functions; and we shall propose to eliminate them by means of the differential equations. Let the primitive equation be

we shall find

and by differentiating again

y +

ly + e + sin x = c,

1 dy

dx y dx

subtracting this from the primitive equation, the functions e

and, it is obvious, that by a new differentiation ly will disappear.
(49.) When m variables are connected together by m-1 equations, any one of them may be considered as
the independent variable, and all the others as implicit functions of it. Hence it may be required to find the
values of the differential coefficients of these implicit functions.

Let u= 0, v = 0, w = 0, &c. be the proposed equations between the variables t, x, y, z, &c., in which t is supposed to be the independent variable, and x, y, z, &c. implicit functions of t. Then u, v, w, &c. may be considered as functions of t, and therefore their first differential coefficients, with respect to that variable, will be respectively (39.)

du dt

du dy
d t

du dx d t

+

+

dy

But the functions of functions of t, denoted by u, v, w, are equal to zero; since by the hypothesis, if we substitute for x, y, z in them, the functions of t they represent, the equations u 0, v = 0, w 0 must be verified. Therefore the differential coefficients of these functions must also be equal to zero. Hence

Part I.

The formation of the equations upon which depends the determination of the differential coefficients
dx
dy
de' dt'
t

de z
dt
d t2

&c. will present no difficulty. It is clear that the differential coefficients of the second order of the functions u, v, w, &c., considered as functions of functions of t, will be obtained by taking the dr dy dz differential coefficients of their first differential coefficients, in which, it must be remembered, dt' dt' di'

&c. are new variables representing functions of t. These differential coefficients of the second order must, as well as those of the first, be equal to zero. Hence will result a sufficient number of equations to calculate the &c., which quantities they will involve in the first degree. That which

d2 y

values of

d to

d t

higher orders, are sufficiently indicated by what precedes, and require no further explanation.

(50.) The observations which have been made before in the case of a single equation between two variables, with respect to the combinations of the primitive equation, and its differential equations, apply clearly here. Between the m - 1 primitive equations, and the m – 1 differential equations of the first order 2 m - 3, constant or variable quantities may be eliminated; and generally between the m 1 primitive equations and n m — n differential equations of the n first orders, (n + 1) m − n − 2 quantities may be eliminated. Let us take, for example, the two equations

And by taking the differential coefficients of the left sides of these equations, we shall have

Either of them may be considered as a function Let us propose to find the values of the partial

(51.) We shall now proceed to examine implicit functions of two or more variables.
Let u=0 be an equation containing three variables x, y, z.
of the other two; z, for instance, as a function of x and y.
differential coefficients of z.

dz

we should

dr

Hence, in the

This will be very easy; for if 2 were expressed by an explicit function of x and y, to determine
consider y as a constant in that function, and then differentiate it as a function of r alone.
present case, we shall first suppose that z and a are the only variables in u, and we shall have by (42)

Secondly. We shall consider x as a constant, and we shall have to determine the value of

will present no difficulty. To find the first we shall

Differential take, as in (43), the differential coefficient of the left side of (a), y being still supposed to be a constant, but
Calculus.

dz

being considered as a variable, we shall have to determine

the equation,

d x2

dr

of the left side of (a) with respect to y, or the differential coefficient of the left side of (b) with respect to .
The two results will be found to be

ď u dz dz du ď z

+
dz2 dx dy dz dxdy

= 0.

No farther explanation is required to understand how the partial differential coefficients of a superior order may
be determined.

(52.) If instead of one equation between three variables x, y, z, we had m equations between m+n variables,
it is obvious that any m of them could be considered as implicit functions of the n remaining.

Let x, y, z, &c. represent the n independent variables, and x', y', z', t', &c. the m variables which are con sidered as functions of them. Each of the m equations may be differentiated in the supposition of a being the d x' dy' dz' only independent variable, and lead to m equations involving the differential coefficients

dť

dx

dx dx dx'
and sufficient to determine their values. The same operation may be repeated on the given equations,
y being then considered as the independent variable, and lead to m new equations, by means of which the
dy' d z'
values of the m differential coefficients
dy dy'

&c. may be obtained. In following the same

dx
dy
process we shall find the mn partial differential coefficients of the first order. It will be sufficient to differen-
tiate the m n preceding equations in a similar manner; and in considering the partial differential coefficients
already involved in them, as new variable functions of x, y, z, to arrive at new equations which will give the
partial differential coefficients of superior order.

These various equations would be verified, as well as the proposed equation, if for x', y', z', &c. the functions
of x, y, z, which they represent, were substituted. Hence they may be combined in any way, and lead to
new equations which will also be satisfied by the same values of x, y, z, &c. Consequently, constant or
variable quantities may be eliminated between them.

The denominations of partial differential equations of the first, second order, &c., and the degree of a partial differential equation of a given order, can be easily understood from what has been said (45), and do not require any further explanation.

The elimination between partial differential equations, presents important results, which we shall now examine.

Let u= 0 be an equation between three variables x, y, z, and let t denote a certain function of x and y, a function of which ƒ(t) s is involved in u. So that if t(x, y), u may be represented by F (s, x, y, z), or by F (ƒ (Þ (x, y)), x, y, z). Hence, if we apply to the equation u = 0 the rules for differentiating functions of functions, we shall have

ds
two quantities s and
may be eliminated. The result will be a partial differential equation of the first
dt
order, not containing s = ƒ (t), and which therefore will be verified by the primitive equation u = 0, whatever
be the form of the function of t designated by f. Thus it appears, that by means of the two partial differential

VOL. I.

5 L

Part 1.